Link Floer Homology and Kleinian Groups

链接是嵌入在三维空间中的一组不相交的圆，它们可以链接在一起。低维拓扑学的主要内容之一是研究链环、三维空间以及由三维空间限定的一些四维空间的拓扑和几何性质。这个项目旨在加深对这些数学结构的理解。研究的第一部分集中于通过所谓的“Dehn手术”运算从链环获得的三-流形以及出现在代数几何中的链环族。这些结果有望促进对代数几何中三维流形和代数奇点的复杂性的理解。它还将提供适合本科生研究的主题。第二部分研究了双曲流形的拓扑、几何和动力学，它们是Gromov双曲空间、负曲Hadamard流形和非紧型对称空间的重要例子。第一个目的是了解在三个球体中的两个组成部分上可能存在的手术障碍。它集中于找到一个整数同调球面的无限族的可能性，这些整形同调球面不能通过对三个球面中的两个分量环进行手术来获得。第二个项目是理解由复平面上代数曲线的奇点产生的代数链环的链Floer链复，并在低维拓扑中提供潜在的应用。第三个项目是研究作用在具有小临界指数的双曲空间上的离散等距子群，并将双曲流形的结构定理推广到负曲Hadamard流形。第四个项目涉及双曲流形中的计数问题，目的是确定经典的Bowen-Marguis度量和谱间隙是否收敛于强收敛的双曲流形序列。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A link is a collection of disjoint circles that may be linked together embedded in a space of dimension three. One of the main topics in low-dimensional topology is to study the topological and geometric properties of the link, the three-dimensional space, and some four-dimensional spaces bounded by the three-space. This project aims to deepen understanding of these mathematical structures. The first part of the research concentrates on the study of three-manifolds obtained from links via the so-called "Dehn surgery" operation and the family of links appearing in algebraic geometry. Results are anticipated to advance the understanding of the complexity of three-manifolds and algebraic singularities in algebraic geometry. It will also provide topics that are suitable for undergraduate students' research. The second part of the research focuses on the topology, geometry, and dynamics of hyperbolic manifolds, which are important examples of Gromov hyperbolic spaces, negatively curved Hadamard manifolds, and symmetric spaces of non-compact type.The research consists of four specific projects about links and hyperbolic manifolds. The first aims to understand the possible obstructions for surgeries on 2-component links in the three-sphere. It focuses on the possibility of finding an infinite family of integer homology spheres that cannot be obtained by surgeries on 2-component links in the three-sphere. The second project is to understand the link Floer chain complex of algebraic links coming from the singularities of algebraic curves in the complex plane and provide potential applications in low dimensional topology. The third project is to study discrete isometry subgroups acting on hyperbolic spaces with small critical exponents and generalize the structure theorem for hyperbolic manifolds to negatively curved Hadamard manifolds. The fourth project concerns a counting question in hyperbolic manifolds, with the goal of determining whether the classical Bowen-Margulis measure and the spectral gap converge for a strongly convergent sequence of hyperbolic manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.